REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

The numbers in brackets are assigned according to the indexing system printedin Volume 22, Number 101, January 1968, page 212.

1[2, 3, 4].—A. César de FREiTAS, Introduçâo à Análise Numérica, Vol. I, Uni-versidade de Lourenço Marques, Moçambique, 1968, xii + 194 pp., 23 cm.Price $5.50.

This is the first of two volumes of an introductory textbook on NumericalAnalysis. It contains the usual material on errors, finite differences, interpolation,numerical differentiation and quadratures, a very short review of numerical integrationof ordinary differential equations, and a fairly extensive (relatively speaking) treatmentof basic numerical linear algebra. This is probably one of the few books on the subjectoriginally written in the Portuguese language and in that sense has special interest.The printing and general presentation are good.

V. PereyraUniversidad Central de VenezuelaCaracas 105, Venezuela.

2[2, 3, 4].—I. P. Mysovskih, Lectures on Numerical Methods, Wolters-NoordhoffPublishing, Groningen, 1969, iii + 344 pp., 23 cm. Price $12.50.

The Russian version of this book appeared in 1962 and the subject matter appearsto predate 1960. This is an excellent book when considered as a mathematical text.However, the author imposes a guideline which the computational scientist will findvery restrictive. This is:

The only computational problems considered seriously are those forwhich it is possible to place a rigorous numerical bound on the accuracyof the result.

For example, to illustrate how to locate a zero, the problem treated is one forwhich it is known that fid)-fib) is negative and f'(x) is positive between a and b.The numerical quadrature section is extensively illustrated by integrating sin x/xbetween 0 and 1. It happens here that the nth derivative is bounded by (n + l)"1.Using this information, one starts by deciding which quadrature rule to use on thebasis of the standard bound on the discretisation error. In the section on numericaldifferentiation, the words 'round-off error' do not occur.

This reviewer is a stranger to the world where numerical problems are so tractable,but I did enjoy reading about it. The book has four chapters. These treat numericalsolution of equations, algebraic interpolation, numerical quadrature and initialvalue ordinary differential equations respectively. In each chapter a few methodsare chosen and are given a careful, clear and rigorous treatment. The section on

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divided differences, for example, succeeded in being thorough without being ex-cessively long. The Hermite osculating polynomial is treated by means of Cauchy'sTheorem. This treatment happened to be new to this reviewer and seems much moresuitable for sufficiently qualified students than the more familiar long-winded dis-cussion. In the quadrature section, the relations between the trapezoidal rules andperiodic integrands is introduced early on and a treatment of Bernoulli functionsand polynomials is included.

I would recommend this book for instructors who will find excellent descriptionsand proofs of various standard theorems in these fields. But students should beexposed to a more realistic view of computing than the one which might be inferredfrom reading this book.

J. N. L.

3 [2, 4, 12].—Ralph H. Pennington, Introductory Computer Methods and NumericalAnalysis, 2nd ed., The Macmillan Co., New York, 1970, xi + 497 pp., 24 cm.Price $10.95.

The first edition of this textbook was published in 1965. (For a review thereof,see Math. Comp., v. 20, 1966, pp. 198-199, RMT 43.) This second edition incorporatesa number of changes, revisions, and modernizations without at the same time alteringthe basic character of the original work. More explicitly than before, the authorstresses that—to paraphrase his words—"he is drawn toward the needs of depart-ments devoted to science, engineering, business administration, and so on", which"find it necessary to include some computer courses for their own purposes". In linewith this, fundamental concepts and rigor "were allowed to suffer to some degreein order to include a sufficient number of descriptions of algorithms to leave thereader with a reasonably versatile beginner's kit of problem-solving tools". Thebackground prerequisites have remained at the integral calculus level.

The overall organization of the book is essentially the same as before, althoughthe arrangement of the material and its subdivision into chapters has been improvedin places. The first five chapters now present the introduction to the fundamentalsof computers and to FORTRAN programming (using ASA standards), and theremaining eight chapters cover the basic numerical methods traditional for this level.

In recognition of the growing importance of interactive computing, a rathernovel change has been the introduction of a FORTRAN variation for remote-terminal operation. Also, the artificial hypothetical machine language used beforehas been modernized and modified as to resemble somewhat that of, say, the IBM 360series. The discussion of several numerical topics has been added or enlarged. Newlyincluded are Chebyshev series, Romberg integration, and in the differential equationschapter, the Euler-Romberg method, and the Adams-Moulton formulas (instead ofMilne's method). The coverage of error propagation, Gaussian quadrature, andthe Runge-Kutta method has been expanded. The abandonment of Sturm sequencesin favor of Graeffe's method (called erroneously Graeff's method in the text andthe index) should be a matter of opinion; rather questionable appears to be thecomplete replacement of Gaussian elimination by the Gauss-Jordan method for thesole stated reason that the latter gives "shorter and more understandable FORTRAN

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subroutines". As before, there are many illustrative examples, and the number ofexercises appears to be increased.

Altogether the second edition has brought the book more in line with the present-day computer environment, and, in balance, it will probably allow it to continueserving the needs to which it was addressed by the author.

Werner C. Rheinboldt

University of MarylandCollege Park, Maryland 20742

4[2, 12].—Alexandra I. Forsythe, Thomas A. Keenan, Elliott 1. Organick &Warren Stenberg, Computer Science: A First Course, John Wiley & Sons, Inc.,New York, 1969, xviii + 553 pp., 24 cm. Price $9.95.

This book is a development of the School Mathematics Study Group (SMSG)text Algorithms, Computation and Mathematics, designed for high school students.It is a carefully written introductory text, covering fully most of the topics suggestedfor Course Bl : Introduction to Computing, of Curriculum 68, the report of theACM Curriculum Committee on Computer Science. In this reviewer's opinion, itis very suitable for a beginning college or junior college course in Computer Science,and is one of the best among the many texts available for courses at this level.

The work is unusual for an elementary text in that it concentrates from the outseton abstract algorithms and flowcharts for them and uses no programming languageother than the author's own "flowchart language." Three supplementary pro-gramming texts are available for FORTRAN, BASIC, and PL/1. This uniquemethod of organizing the text allows the authors to concentrate on the essentialideas of computing and algorithms unencumbered by the technical details of aparticular language, which the neophyte often finds the most difficult to learn. It alsohas the advantage of allowing the instructor to choose whichever programminglanguage is available to him or which he considers best for beginning students.

The book is divided into three parts. Part I introduces the student to computingand covers algorithms and flowcharts. Part II covers elementary numerical analysisand applications to computing including quadrature, simultaneous linear equations,and linear regression. Part III covers some of the newer areas of Computer Scienceincluding trees, lists, strings, and compiling. A second version of the text, ComputerScience: A Primer, contains Parts I and II only and may be more suitable for highschool use. Also published with the text is Computer Science: Teacher's Commentary,a very detailed supplement which seems intended for mathematics teachers with noprevious experience in Computer Science. It contains complete flowcharts withcomments for all of the problems in the text and additional problems, explanationsand suggestions for the teacher.

The book is mathematically oriented and pays particular attention to carefuldevelopment of mathematical concepts. It is well coordinated with the SMSG mathe-matics texts.

This reviewer has only a few complaints about the book. One is the violationof the ANSI standard for flowcharts, somewhat disturbing in a book with over

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300 flowcharts. Another is a rather strange use of the word 'round' to mean bothround and truncate. The authors assume that the reader is familiar with binaryarithmetic, not necessarily true in this reviewer's experience with college students.The explanation of computer hardware, especially on core storage, is confusingand sketchy. The book has several misprints which may confuse the beginner. How-ever, these are all minor and should be corrected in the second printing.

The exercises are ample and excellent and should serve students with a widerange of aptitudes and interests. In combination with one of the programmingsupplements, or with any programming text, the book should be very successfulin classroom use.

Emile C. ChiCourant Institute of Mathematical SciencesNew York UniversityNew York, New York 10012

5[3].—Robert T. Gregory & David L. Karney, A Collection of Matrices for TestingComputational Algorithms, John Wiley & Sons, Inc., 1969, ix + 154 pp., 28 cm.Price $9.95.

A much needed collection of matrices for testing algorithms, which arise innumerical linear algebra, is provided by this book. The authors provide both well-conditioned and ill-conditioned test matrices for algorithms concerning: (1) inverses,systems of linear equations, determinants and (2) eigensystems of real symmetric,real nonsymmetric, complex, and tridiagonal matrices. The construction of testmatrices is discussed, and a large number of references and a table of symbols isprovided.

The authors do not discuss the perplexing problem concerning how a user mustchoose the appropriate test matrices. In particular, test matrices must be chosenso that all parts of the algorithm are tested. It may not be clear to a user by lookingat the examples which matrix (if any) will go through a particular part of his algorithm.Then, he must construct his own examples by working backwards through hisalgorithm.

James R. BunchUniversity of ChicagoChicago, Illinois 60637

6 [5].—Dale U. Von Rosenberg, Methods for the Numerical Solution of PartialDifferential Equations, American Elsevier Publishing Co., Inc., New York, 1969,xii + 128 pp. Price $9.50.

This book serves as a good introduction for anyone interested in finite differencemethods. In the preface, the author states "This book is written so that a seniorundergraduate or first-year graduate student in engineering or science can learnto use these methods in a single semester course, and so that an engineer in industrycan learn them by self-study." The book succeeds admirably. The style is very readable

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and the physical background of each equation is discussed. Each chapter ends witha specific problem which is completely worked out including a program and computertime.

Specifically, in chapter one, the example of heat conduction in an insulatedtapered rod with various boundary conditions is used to illustrate some methodsfor linear ordinary differential equations.

In chapter two, linear parabolic partial differential equations are treated. Theforward difference, backward difference and Crank-Nicolson schemes are derivedand the stability analysis for each is carried out.

In chapter three linear hyperbolic equations, including systems, are dealt with.Chapter four is concerned with alternate forms of the coefficient matrices generated

by the difference schemes.Chapters five and six deal with nonlinear parabolic and hyperbolic equations,

and chapter seven describes nonlinear boundary conditions.Chapters two through seven deal with equations with one space variable.Chapter eight describes parabolic and elliptic equations in two and three space

dimensions and includes Alternating-Direction-Implicit schemes.Chapter nine mentions some examples of complications which can arise in the

problems treated earlier, for example shock waves in hyperbolic equations.There is an appendix of algorithms useful in solving the types of equations gen-

erated by the difference schemes.One slight drawback for use as a text is that no exercises are provided. More

serious is the fact that function space approximation methods are not mentionedat all, and the title of the book might lead a student to believe that finite differencesare the only known methods for obtaining numerical solutions.

Stephen HilbertIthaca CollegeIthaca, New York 14850

7 [7].—J. W. Wrench, Jr., The Converging Factor for the Modified Bessel Functionof the Second Kind, NSRDC Report 3268, Naval Ship Research and DevelopmentCenter, Washington, D. C, January 1970, ii + 56 pp., 26 cm.

This report, which follows the pattern of two earlier reports [1] and [2] (see Math.Comp., v. 20, 1966, pp. 457-458, RMT 71), is concerned with the development ofmethods and provision of auxiliary tables for the high-precision calculation of theconverging factor in the asymptotic expansion of the modified Bessel function ofthe second kind, Kp{x). While much of the analytical discussion is quite general,the practical application is restricted to functions of integer order and positive realargument.

The converging factor is defined as "that factor by which the last term of a trun-cated series (usually asymptotic) approximating the function must be multipliedto compensate for the omitted terms." Thus in the case of the function K„(x) we have

v t \ tit *'* -41 _l a'(p) _i_ fl2(p) , , aAp) , AK„(x) = ix/2x) e y -\-1-2— + • • • H-— ',,,W| ,

192 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

where the a¡ip) are known coefficients and o-„,p(x) is the converging factor. This mayitself be expanded in a series of the form

REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 193

There are four chapters and five appendices, the titles of which (translated intoEnglish) are: I, "The exact number of primes less than a given limit." II, "The approxi-mate number of primes." Ill, "How to identify large primes." IV, "Factorization."1, "Some algebraic theorems." 2, "Some number theoretic theorems." 3, "Quadraticresidues." 4, "Arithmetic of quadratic fields." 5, "Algebraic factors of a" ± bn."

Chapter I deals in such topics as Meissel's formula and Lehmer's formula. ChapterII discusses distribution questions: the Gauss, Legendre, and Riemann approxima-tions; the remainder term in the prime number theorem; twins, large gaps, etc. Thelast two chapters are concerned with primality tests and factorization methods. Theappendices have background material.

There are 12 extensive tables.1. All primes/; < 104.2. All primes 10" < p < 10" + 1000 for n = 4(1)15.3. 7r(x) and the Gauss and Riemann approximations to x = 10'8.4. Composites 104395289 < c < 2-108, satisfying 2""1 = 1 (mod c) and having

no prime divisor g 317.5. Known factors of Fermât numbers.6. Complete factorization of 2" - 1. AU n < 137 and others g 540.7. Factors of (10" - l)/9 and 10" + 1.8. Primes of form h-2n + 1 for h = 1(2)99. [Note, h = 1, n = 8 is missing.]9. Primes of form «-2" - 1 for h = 1(2)151.

10. Primes of form «4 + 1 for « g 4002.11. Miscellaneous.12. Quadratic residues: for square-free a, \a\ < 100, all k and / such that primes

p = kx + 1 have (a | p) = +1.An English translation may be published in the future, but even readers with no

Swedish will be able to grasp much of the present text. Try this:

SATS B1.3 Om A är element i en ändlig grupp G med n element,är An = I.

The tables are in Arabic numerals and will cause no difficulty at all.D. S.

9 [10].—John Leech, Editor, Computational Problems in Abstract Algebra, PergamonPress, Ltd., Oxford, 1970, x + 402 pp., 23 cm. Price $18.50.

This book consists of thirty-five papers, most of which were presented at a con-ference on the use of computers in solving problems in algebra, held in 1967 at theUniversity of Oxford under the auspices of the Science Research Council AtlasComputer Laboratory.

Over one-half of the papers are concerned with the application of computers toproblems in group theory. The balance of the book includes papers on the use ofcomputers in such areas as word problems in universal algebras, nonassociativealgebras, latin squares, Galois theory, knot theory, algebraic number theory, algebraictopology and linear algebra. The first paper, a survey of the methods used and the

194 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

results obtained by computers in the investigation of groups, contains an extensivebibliography on applications of computers to problems in algebra.

Many of the leading experts in the use of computers in algebra are among theauthors of these papers, and the book gives a fairly comprehensive picture of thequite diverse activity in the subject.

The number of available papers concerning applications of computers to problemsin algebra is relatively meager considering the potential of such applications. Thisvolume is a welcome and a significant addition to the literature on the subject.

Harold BrownThe Ohio State UniversityColumbus, Ohio 43210

10[12].—R. D. Parslow, R. W. Prowse & R. E. Green, Editors, Computer Graphics:Techniques and Applications, Plenum Press, London and New York, 1969, xiv +247 pp., 23 cm. Price $16.00.

This book consists primarily of a collection of papers that were presented at theInternational Computer Graphics Symposium held in July 1969 at Brunei University,Uxbridge, England. The book, which was designed to cover the field of computergraphics, is organized into four parts.

Part 1 contains nine papers that deal with basic hardware/software concepts incomputer graphics. This section of the book is intended to serve as an introduction tothe field for the novice; unfortunately, only the first paper, "What has ComputerGraphics to Offer?", is suitable for this purpose.

The paper "Computer Graphics Hardware Techniques," although well written,suffers from a lack of visual aids to guide the novice. The companion paper "ComputerGraphics Software Techniques" provides a rather sketchy coverage of its subjectmatter. The highlight of this paper is an extremely good treatment of the concept of agraphic data structure and its relationship to a display file. A third techniques-orientedpaper, "Interactive Software Techniques," suffers because of poor organization.

The paper entitled "Computer Display System Tradeoffs" provides an interestingdiscussion of the relative merits of the so-called buffered display, which contains itsown refresh memory, and a display which uses the computer memory as the refreshmemory. This discussion concentrates on the hardware aspects of the question andignores the software considerations. This paper, although well written, may be loston the novice.

The two papers "Computer Graphics in the United States" and "The U. K.Scene" are intended to provide an overview of computer graphics in the United Statesand in England. The poor presentation in the second paper is in very striking contrastto the better organized presentation in the first.

The paper "Low Cost Graphics" provides a very good coverage of the tradeoffsinherent in the three most commonly used types of CRT displays; random-scanrefresh, sequential-scan refresh, and the direct-view storage tube (DVST). A strongcase was made for the use of the DVST in those applications which are not highlyinteractive.

REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 195

The final paper in Part 1, entitled "Remote Display Terminals," deals with thevarious types of display terminals suitable for use in a remote on-line interactiveenvironment. The so-called low-speed terminals (e.g., teletype, incremental plotters,and alphanumeric CRT's) receive the bulk of the attention, while CRT's providingthe full range of graphics receive very sketchy treatment.

Part 2 contains nine papers, eight of which describe graphic applications. Thegraphical discussions contained in these eight papers are, for the most part, not of ageneral nature and hence would be of little interest to anyone not familiar with therather specialized applications covered. The final papers in this section consist of acollection of remarks and observations which were recorded during a discussionsession at the Symposium.

There is only one paper in Part 3. This paper, "Present Day Computer GraphicsResearch," gives a good coverage of research efforts in low-cost terminals and ingraphic software.

The reference section of the book, Part 4, consists of a series of advertisementsfor hardware manufacturers of graphics equipment, a glossary of computer graphicsterms, and a consolidated bibliography for all of the papers appearing in the book.The glossary provides a comprehensive coverage of graphics-related terms, and isperhaps the most useful portion of the book.

Allen L. Hankinson

Department of Applied MathematicsNaval Ship Research and Development CenterWashington, D. C. 20034

11 [12].—A. Van Wijngaarden, Editor, B. J. Mailloux, J. E. L. Peck & C. H. A.Koster, Report on the Algorithmic Language ALGOL 68, Second Printing MR101, Mathematisch Centrum, Amsterdam, 1969, v + 134 pp., 24 cm. Price $4.50.Also available as offprint from Numerische Mathematik, v. 14, 1969, pp. 79-218,Springer-Verlag, New York.

The report is the culmination of a five-year effort by Working Group 2.1 of theInternational Federation for Information Processing (IFIP) to design a successor toALGOL 60. Section 0 of the report describes the aims and principles of design, thefirst of which is "completeness and clarity of description." By Section 1, the clarityhas disappeared. Embedded among the specifications of syntax and semantics arecomments, called pragmatics, intended "to help the reader understand the implica-tions of the definitions." However, they are often little more than verbalization ofthe syntax rules. The reader well-versed in the language may find the report usefulas an authoritative reference manual (provided he has learned all the new terminology),but even the reader experienced in programming languages is advised to seek otherexpositions to learn the language. (Such documents are beginning to be available incomputer science literature.)

As for the language, although it is not strictly an extension of ALGOL 60, it isin the same tradition. (A brief comparison of ALGOL 60 and ALGOL 68 is given inSection 0 of the report.) The ALGOL 60 notion of type is generalized to the concept

196 REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS

of mode, of which there are an infinite variety and which include structured valuesand names (pointers, here called "references"). Subscripting is generalized to includeslices. There is a more general conditional statement. (Statements are called "clauses"in ALGOL 68.) The language has facilities for formatted input-output (called "trans-put") and for environment inquiries.

The opinion has been expressed (cf. Minority Report, ALGOL Bulletin 31, p. 7)that ALGOL 68 is not sufficiently advanced to facilitate the reliable creation of today'smore sophisticated programs. Since no implementation of the language has beencompleted (although several efforts are well advanced), experience is not availableto confirm or deny this view.

The offprint from Numerische Mathematik has exactly the same material as theMathematisch Centrum edition but is much easier on the eye because of better choicesof type and better spacing.

Susan L. Graham

Courant Institute of Mathematical SciencesNew York UniversityNew York, New York 10012

12 [13.35].—Arto Salomaa, Theory of Automata, International Series of Monographsin Pure and Applied Mathematics, Vol. 100, Pergamon Press Ltd., Oxford, 1969,xii + 263 pp., 22 cm. Price $12.00.

This is a well organized introduction to finite automata theory. It deals with themathematical foundations, and not with the practical applications to sequentialswitching circuits or nerve networks. The construction and programming of actualcomputers is not treated. Instead, the "machines" considered are certain theoreticalmodels which have been intensively investigated during the last fifteen years. Theseinclude the finite deterministic automaton, the finite nondeterministic and probabilisticautomata, and the pushdown and linear bounded automata.

Attention centers on the languages which are representable in the various machines.Regular languages (those representable in finite deterministic automata) and stochasticlanguages (those representable in probabilistic automata) are studied in the first twochapters. The third chapter is devoted to the algebra of regular expressions; herethe author presents some of his own results concerning axiomatizations of this algebra.

The fourth and last chapter, entitled "Formal languages and generalized auto-mata," introduces the notion of generation of languages by grammars. It includesproofs that the context-free languages are those which are representable in non-deterministic pushdown automata and that the context-sensitive languages are thosewhich are representable in nondeterministic linear bounded automata. Recursivelyenumerable sets and Turing machines are mentioned in this chapter; however, thefull study of recursive function theory is outside the scope of the book. This chapteralso includes a section on the abstract pushdown automata of Letichevskiï. Theauthor has prepared the reader for this section by including earlier sections on theanalysis of finite automata by means of characteristic equations and on the solution

REVIEWS AND DESCRIPTIONS OF TABLES AND BOOKS 197

of equation systems in which the unknowns are languages. There are many otheritems of special interest which cannot be mentioned in a brief review.

Algebraic structure theories, such as those developed by Krohn and Rhodes, byHartmanis and Steams, and by Eilenberg and Wright, are not considered in this book.The author has, however, provided a collection of historical and bibliographicalremarks, which, together with an extensive list of references, make the book a usefulguide to the literature on automata theory, in both the English and Russian languages.Many exercises which explain and extend the theory are listed, and some statementsof unsolved problems are given as well, so that the book can be conveniently usedas a text in a seminar or in a beginning graduate course.

George N. Raney

University of ConnecticutStorrs, Connecticut 06268